Check-weight-constrained quantum codes: Bounds and examples

TL;DR

This work establishes fundamental limits for quantum LDPC codes under strict check-weight constraints, proving that stabilizer codes with weight-3 checks cannot achieve nontrivial distance while CSS and subsystem variants with weight-4 and weight-2 checks obey tight $k$–$d$ tradeoffs (e.g., $kd^2=O(n)$ and $d\le\sqrt{n}$, $kd\le n$). The authors derive finite-size upper bounds using linear programming that incorporate MacWilliams identities and check-weight constraints, and they identify explicit quantum Tanner-code constructions that approach these bounds for tens to hundreds of qubits. Unlike locality-dependent results, these bounds apply to generic qLDPC codes, revealing inherent tradeoffs between check weight and code parameters absent geometric locality. In addition to asymptotic bounds, the paper provides numerical LP frontiers, practical code constructions, and open-source parity-check matrices to guide near-term experimental demonstrations of quantum error correction with constrained checks.

Abstract

Quantum low-density parity-check (qLDPC) codes can be implemented by measuring only low-weight checks, making them compatible with noisy quantum hardware and central to the quest to build noise-resilient quantum computers. A fundamental open question is how constraints on check weight limit the achievable parameters of qLDPC codes. Here, we study stabilizer and subsystem codes with constrained check weight, combining analytical arguments with numerical optimization to establish strong upper bounds on their parameters. We show that stabilizer codes with checks of weight at most three cannot have nontrivial distance. We also prove tight tradeoffs between rate and distance for broad families of CSS stabilizer and subsystem codes with checks of weight at most four and two, respectively. Notably, our bounds are applicable to general qLDPC codes, as they rely only on check-weight constraints without assuming geometric locality or special graph connectivity. In the finite-size regime, we derive numerical upper bounds using linear programming techniques and identify explicit code constructions that approach these limits, delineating the landscape of practically relevant qLDPC codes with tens or hundreds of physical qubits.

Figures (13)

• Figure 1: (a) Visualization of the upper bounds on code parameters of CSS codes in the $(n,k,d)$ space as a function of check weight $w$ in different colors, where $n$ and $k$ are the number of physical and logical qubits, respectively, and $d$ is the distance. (b) Parameters of our quantum Tanner codes as well as other explicit qLDPC codes in the literature with check weight $w\le 10$. The solid curves are the LP achievability frontiers, i.e., the maximum rates $R_{LP}(\delta)=\max\{R_i:\delta_i\geq\delta\}$ over all feasible points of at least the given relative distance (considering bounds obtained for $n\le 300$) and fixed $w$.
• Figure 2: Proof of Lemma \ref{['lem:wt4homologicalcode']}. (a) The code $\mathcal{C}$ is placed on a graph with the qubits associated with the different colored edges. (b) The $Z$ checks are the qubits along the cycles $123$, $234$, $134$, and $124$. (c) By gluing the different cycles together in the correct orientation where they share the same edge, we obtain a closed surface (a sphere in this case) on which $\mathcal{C}$ is a generalized surface code.
• Figure 3: A log-log plot visualization of bounds on subsystem codes with $n$ physical qubits and check weight 2. The green region corresponds to the achievable parameters $(k,d)$, where $k$ is the number of logical qubits and $d$ is the distance. Codes in the green region can be realized by the construction in Ref. bravyi2011subsystem. The parameters in the red region are ruled out by Theorem \ref{['thm:subsystem']}.
• Figure 4: (a) An example of a subsystem code with check weight 2. Red (blue) lines and circles indicate $X$ ($Z$) checks of weight two and one, respectively. (b) The associated matrix $A$, where the rows correspond to the weight-2 $X$ components $\{X_2, X_3\}$, $\{X_4, X_6\}$, $\{X_7, X_8, X_9\}$ and the columns correspond to the weight-2 $Z$ components $\{Z_1, Z_4, Z_7\}$, $\{Z_2, Z_5, Z_8\}$, $\{Z_3, Z_6\}$. Entries correspond to the parities of the number of common qubits between the components.
• Figure 5: Cross-sections of upper bounds on code parameters of CSS codes from Fig. \ref{['fig:CSSall3D']}(a), with the different colors representing bounds for different check weights. We pick a representative value of $n$, $k$, or $d$ to fix in (a), (b), and (c), respectively. Parameters above the points in (a) and (c), and those in (b) above or to the left of the furthest point of a given check weight, are ruled out by the LP.

Theorems & Definitions (39)

• Definition 2.1: disentangled subsystem
• Definition 3.1
• Definition 3.2: minimal code
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6